2. Magnetic islands in a perturbative approach

Magnetic islands are due to non-vanishing resonant magnetic perturbations in the plasma, and a perturbative approach is therefore frequently used for their description. In particular, a magnetic island with poloidal $m$ and toroidal $n$ periodicity opens around the resonant flux surface defined by the rational value $\iota =n/m$ of the rotational transform related to the unperturbed equilibrium configuration.

The island width is commonly computed in terms of the geometric width of a pendulum-like cat's eye in the context of small resonant perturbations of a regular magnetic field, associated with a time-independent Hamiltonian (Hazeltine & Meiss Reference Hazeltine and Meiss1992; White Reference White2013). In all cases, the width of a magnetic island is proportional to the square root of a magnetic flux, which turns out to be the perturbation of a helical flux evaluated on the resonant flux surface (Park et al. Reference Park, Boozer and Menard2008; Predebon et al. Reference Predebon, Momo, Terranova and Innocente2016). As shown in § 3, this flux can be interpreted as the helical flux through the separatrix.

In this section, we exploit the existence of a coordinate-independent magnetic flux related to a magnetic island that correctly estimates its width, as proved by Escande & Momo (Reference Escande and Momo2024): this flux, named $\Phi _{{O}{X}}$, is defined for each magnetic island through the ribbon defined by the periodic orbits related to the $O$- and $X$-points shown in figure 1. The above path is based on the variational principle formulation for magnetic field lines and on the related action

(2.1)\begin{equation} S_\gamma = \int_{\gamma} \boldsymbol{A}(\boldsymbol{x}) \boldsymbol{\cdot} {\mathrm{d}}\kern0.7pt \boldsymbol{x}, \end{equation}

where $\boldsymbol {x}$ is the spatial coordinates vector, $\boldsymbol {A}$ is the vector potential, $\boldsymbol {B} = \boldsymbol {\nabla } \times \boldsymbol {A}$ and the integral runs along the path $\gamma$ between two points of a magnetic field line. When $\gamma$ is a closed circuit, the Stokes theorem states that the action $S_\gamma$ is the magnetic flux through the surface having this circuit as a boundary.

Figure 1. Helical ribbon defining the $\Phi _{{O}{X}}$ flux for a $(1,1)$ magnetic island, with the closed orbits corresponding to the $O$- and the $X$-point in black and red, respectively.

To relate the action $S_\gamma$ to the resonant perturbation that opens an island, the formal identification between canonical and magnetic coordinates, and the equivalence between the covariant components $A_i, i=1,2,3$ (where the index $i$ corresponds e.g. to the radial, poloidal and toroidal coordinate, respectively) of the vector potential and the magnetic fluxes must be used in the definition of $S_\gamma$. This equivalence is only valid in the axial gauge $A_1=0$,

(2.2)\begin{align} S_\gamma & = \int_{\gamma} (A_2 \,{\mathrm{d}} x^2 + A_3 \,{\mathrm{d}} x^3) \nonumber\\ & = \frac{1}{2 {\rm \pi}} \int_{\gamma} (p \,{\mathrm{d}} \vartheta - H \,{\mathrm{d}} \varphi) \nonumber\\ & = \frac{1}{2 {\rm \pi}} \int_{\gamma} (\psi_t \,{\mathrm{d}} \vartheta - \psi_p \,{\mathrm{d}} \varphi), \end{align}

with $x^2 = \vartheta$ and $x^3 = \varphi$ poloidal and toroidal angles; the relations $p=\psi _t$ and $H=\psi _p$ define the identifications between the canonical momentum and the toroidal magnetic flux, and between the Hamiltonian and the poloidal flux, respectively, assuming the canonical position and time to be $q=x^2=\vartheta$ and $t=x^3=\varphi$.

Figure 1 visualizes the helical ribbon defined by the orbits of the $O$- and $X$-points, and helps understand the geometrical meaning of the flux $\Phi _{{O}{X}}$ through that ribbon. Using the definition of the action for a magnetic field line, the $\Phi _{{O}{X}}$ flux turns out to be $S_{O}-S_{X}$, where $S_{O}$ is the action computed along the closed orbit defined by the $O$-point, and $S_{X}$ the action along the closed orbit defined by the $X$-point. In the rest of this section, we revisit the derivation of § 5 of the review by Escande & Momo (Reference Escande and Momo2024) writing the island width as a function of $\Phi _{{O}{X}}$.

Let $\boldsymbol {x}= (\psi _t^0,\vartheta,\varphi )$ be magnetic coordinates for the unperturbed equilibrium. If the perturbation is not large enough to violate the requirement of a non-null Jacobian, then the full perturbed system around the resonant surface, in the same $\boldsymbol {x}$ coordinate system, is approximated by

(2.3)\begin{gather} \psi_t(\boldsymbol{x}) \simeq \psi_t^0 + \psi_t^{m,n}(\psi_t^0) \, {\rm e}^{{\rm i} u} + {\rm c.c.}, \end{gather}

(2.4)\begin{gather}\psi_p(\boldsymbol{x}) \simeq \psi_p^0 + \psi_p^{m,n}(\psi_t^0) \, {\rm e}^{{\rm i} u} + {\rm c.c.}, \end{gather}

where $u=m \vartheta -n \varphi$ is called helical angle and ${\rm c.c.}$ indicates the complex conjugation. The unperturbed flux, $\psi _p^0=\psi _p^0(\psi _t^0)$, defines the unperturbed equilibrium and its flux surfaces through the relation $\iota ={\mathrm {d}} \psi _p^0/{\mathrm {d}} \psi _t^0={\mathrm {d}} \vartheta / {\mathrm {d}} \varphi$. The $\psi ^{m,n}(\psi _t^0)=|\psi ^{m,n}| \,{\rm e}^{{\rm i} \alpha ^{m,n}}$ terms are the Fourier components of the fluxes having the resonant periodicity.

We now better define the actions $S_{O}$ and $S_{X}$, which are the line integrals along the lines defined by the $O$- and $X$-point of the $(m, n)$ magnetic island, respectively. In calculating $S_{O}$ and $S_{X}$ from (2.2) using definitions (2.3) and (2.4), we assume that $m$ and $n$ are mutually relatively prime; $\varphi$ varies by $2{\rm \pi} m$ along $O$ or $X$ and $\vartheta$ varies by $2{\rm \pi} n$, while the helical angles along the $O$ and $X$ orbits ($u_O$ and $u_X$, respectively) are constant.

We first compute the action $S_{O}$:

(2.5)\begin{align} S_{O} & = \frac{1}{2 {\rm \pi}} \int_O (\psi_t {\mathrm{d}} \vartheta - \psi_p {\mathrm{d}} \varphi) \nonumber\\ & = ( n \psi_t^0 - m \psi_p^0 )|_{res} + ( n \psi_t^{m,n} - m \psi_p^{m,n})|_{res} \, {\rm e}^{{\rm i} u_O} + {\rm c.c.}, \end{align}

where all radial functions must be evaluated on the rational surface defined by $\iota =n/m$, even if not explicitly stated in the following notation. Introducing the helical flux function

(2.6)\begin{equation} \psi_h(\psi_t^0, u)= m \psi_p - n \psi_t , \end{equation}

$S_{O}$ can be written in terms of this flux through the $O$-point orbit as

(2.7)\begin{align} -S_{O} & = \psi_h(\psi_t^0, u_O) \nonumber\\ & = \psi_h^0 + |\psi_h^{m,n}| \, {\rm e}^{{\rm i} (u_O+\alpha_h^{m,n})} + {\rm c.c.}, \end{align}

where $\psi _h^0=\psi _h^0(\psi _t^0)$ is the unperturbed equilibrium flux, whereas $| \psi _h^{m,n} |$ and $\alpha _h^{m,n}$ are the amplitude and phase of the $(m,n)$ Fourier component.

The calculation of the $S_{X}$ flux follows the same steps, with $u_O$ being substituted by $u_X$, and remembering that $u_X$ is shifted by ${\rm \pi}$ with respect to $u_O$: $u_O+\alpha _h^{m,n}=0, {\rm \pi}$ and $u_X+\alpha _h^{m,n}={\rm \pi},0$, depending on the sign of the magnetic shear. This yields

(2.8)\begin{align} -S_{O}(\psi_t^0) & = \psi_h^0 + 2 | \psi_h^{m,n} | \cos(u_O+\alpha_h^{m,n}) \nonumber\\ & = \psi_h^0 \pm 2 | \psi_h^{m,n} |, \end{align}

(2.9)\begin{align} -S_{X}(\psi_t^0) & = \psi_h^0 + 2 | \psi_h^{m,n} | \cos(u_X+\alpha_h^{m,n}) \nonumber\\ & = \psi_h^0 \mp 2 | \psi_h^{m,n} |, \end{align}

(2.10)\begin{align} \Phi_{{O}{X}} & \equiv S_{O}-S_{X} ={\mp} 4 | \psi_h^{m,n} |, \end{align}

with all radial functions evaluated on the rational surface. The minus sign in (2.10) corresponds to a negative magnetic shear at the rational surface (${\mathrm {d}} \iota /{\mathrm {d}} \psi _t^0 < 0$), which implies $u_O+\alpha _h^{m,n}=0$, while the positive sign corresponds to the opposite case (${{\mathrm {d}} \iota /{\mathrm {d}} \psi _t^0 > 0}$), with $u_O+\alpha _h^{m,n}={\rm \pi}$.

From an operative point of view, the $\Phi _{{O}{X}}$ flux can be computed both from (2.10) or by solving numerically the line integrals in (2.2) for $\gamma =O$ and $\gamma =X$. In the first case, one needs to evaluate the helical flux perturbation at the resonant surface; in the second case, one needs to know the path of the island extrema.

The amplitude of the magnetic island can then be computed from the formula (similar to (90) of Escande & Momo Reference Escande and Momo2024)

(2.11)\begin{equation} W_{\Phi_{{O}{X}}} = 4 \sqrt{\left| \frac{\Phi_{{O}{X}}}{2m \dfrac{ {\mathrm{d}} \iota}{{\mathrm{d}} \psi_t^0}} \right|} \left(\frac{{\mathrm{d}} r}{{\mathrm{d}} \psi_t^0}\right),\end{equation}

where the factor $({\mathrm {d}} \psi _t^0/ {\mathrm {d}} r)^{-1}$ brings the width in units of a length. It implies a clear relation between the unperturbed flux $\psi _t^0$ and a radial coordinate $r$ in meters: when $r$ is the radius of the circular flux surface's cross-section of the zeroth-order equilibrium, the factor $({\mathrm {d}} \psi _t^0/ {\mathrm {d}} r)^{-1}$ considers the same amplitude at any poloidal or toroidal cuts. All radial functions, as the magnetic shear or the $({\mathrm {d}} \psi _t^0/ {\mathrm {d}} r)^{-1}$ term, must be evaluated at the resonant surfaces.

3. Magnetic islands as new equilibrium systems

In § 2, we stated the equivalence between the magnetic action along a helical path $\gamma$ and the helical flux. From this perspective, (2.8) and (2.9) can be written in a shorter notation:

(3.1)\begin{gather} - S_{O} = \psi_h|_O, \end{gather}

(3.2)\begin{gather}- S_{X} = \psi_h|_X, \end{gather}

and therefore,

(3.3)\begin{equation} \Phi_{{O}{X}} \equiv S_{O}-S_{X} ={-} \psi_h|_O + \psi_h|_X, \end{equation}

where $\psi _h|_O$ means the helical flux evaluated along the line defined by the $O$-point, and similarly $\psi _h|_X$. From a geometrical point of view, $S_{O}$ can be interpreted as the helical flux $\psi _h|_O$ through the the surface delimited by the orbit of the centre of the island ($O$-point), and $S_{X}$ as the helical flux $\psi _h|_X$ through the edge of the island (the orbit defined by the $X$-point). In this section, as well as in § 4, a way to compute the magnetic fluxes through any island flux surface is shown.

The perturbative approach in § 2 assumes an integrable unperturbed magnetic field configuration, i.e. the equation of motion can be solved to give non-chaotic magnetic field lines and therefore conserved magnetic flux surfaces. In the presence of general magnetic perturbations, the system is not integrable and flux surfaces are destroyed. Apart from an axisymmentric system, there is only one other known integrable system, i.e. that of helical symmetry, that defines conserved magnetic flux surfaces, analogous to the constant energy surfaces. In both cases, the Hamiltonian is time-independent and the equivalence $t=\varphi$ holds.

Equations (2.3) and (2.4), adding to the unperturbed equilibrium a single Fourier perturbation, define a helical integrable Hamiltonian. To integrate it, we make use of the change of coordinates $(\psi _t^0,\vartheta,\varphi )\mapsto (\psi _t^0,u,\varphi )$, where $u=m \vartheta - n \varphi$ is the helical variable. This change of coordinates defines a new time-independent Hamiltonian: the helical flux $\psi _h(\psi _t^0,u)= m \psi _p-n \psi _t$, which can be assumed as a radial variable of any system with a helical symmetry (Hazeltine & Meiss Reference Hazeltine and Meiss1992). In fact, in the new coordinates, the identifications with the $(p,q,t)$ variables are $q=x^2=u, t=x^3=\varphi$ and therefore $p=\psi _t, H=\psi _h$.

The island domain can be modelled as a helical equilibrium configuration, and a reconstruction of such equilibria has been implemented in a code named SHEq (Martines et al. Reference Martines, Lorenzini, Momo, Terranova, Zanca, Alfier, Bonomo, Canton, Fassina, Franz and Innocente2011), now extended to the tokamak case too. The method is based on the superposition of an axisymmetric equilibrium and of a first-order helical perturbation computed according to Newcomb's equation supplemented with edge magnetic field measurements (Zanca & Terranova Reference Zanca and Terranova2004); more details are in § 5. The helical flux contours give the shape of the flux surfaces of a helical domain. An example of such surfaces is shown in figure 2 (bottom half of panel a) for a magnetic island in an RFP plasma. The accuracy of the flux surface reconstruction is confirmed by the corresponding discretized field map (top half of panel a) obtained with the field line tracing code Flit, which integrates the field lines with the same helical Fourier perturbations (Innocente et al. Reference Innocente, Lorenzini, Terranova and Zanca2017). The value of helical flux through the surfaces delimited by the $X$- and $O$-point orbits (their intersection with the poloidal plane respectively plotted in the figure with an orange and green dot) can be identified from the value of the helical flux at the extrema of its profile on the equatorial plane (figure 2b). The values of the helical flux above the value at the separatrix (orange dashed line in figure 2b) label the external flux surfaces with respect to the magnetic island, so the helical flux profile computed by the SHEq code is here cut at the separatrix to restrict the computation to the island domain. The values of $\psi _h$ in the well between the $X$- and the $O$-point correspond to the island flux surfaces; whereas the values of $\psi _h$ in the other well correspond to the circular flux surfaces around the axisymmetric equilibrium axis. It is worth noting that the evaluation of the helical flux on the $X$- and $O$-points enables an evaluation of the $\Phi _{{O}{X}}$ flux from (3.3) and, therefore, the island width from (2.11).

Figure 2. (a) $\varphi =0$ discretized field map of a ($1,7$) island (top half of the section) and corresponding flux surface contour (bottom half) in a circular RFP. Red and light-blue curves in the top half-panel represent the two last flux surfaces from the Flit code (Innocente et al. Reference Innocente, Lorenzini, Terranova and Zanca2017), the thick line in the bottom half-panel the separatrix computed by the Sheq code (Martines et al. Reference Martines, Lorenzini, Momo, Terranova, Zanca, Alfier, Bonomo, Canton, Fassina, Franz and Innocente2011), and coloured dots the $X$- (orange) and $O$-points (green). (b) Helical flux $\psi _h$ on the equatorial plane passing through the $X$- (orange) and $O$-points (green).

We can now define a new reference frame having its axis on the $O$-point of the island. From here on, we introduce the new index $\cdot _H$ to explicitly identify the quantities related to the helical magnetic flux surfaces $\varSigma _H$ from the $O$-point to the separatrix. A time-independent canonical change of coordinates allows us to write the Hamiltonian of the helical system, $\psi _h$, in its action-angle form, ${\psi _{H}}$, where both the poloidal and toroidal fluxes across the helical flux surfaces are constants of the motion (Momo et al. Reference Momo, Martines, Escande and Gobbin2011) and therefore functions of ${\psi _{H}}$. Due to the time independence of the canonical transformation, $\psi _h$ and ${\psi _{H}}$ define the same flux surfaces. Using the definition of the canonical action coordinate, the identifications between canonical and magnetic coordinates, and the perturbed fluxes in (2.3) and (2.4), the toroidal flux through $\varSigma _H$ is defined by

(3.4)\begin{align} {\psi_{{H} t}}({\psi_{H}}) & = \frac{1}{2 {\rm \pi}} \oint p(E,q) \, {\mathrm{d}} q = I(E) \end{align}

(3.5)\begin{align} & = \frac{1}{2 {\rm \pi}} \oint_{\varSigma_H} \psi_t(\psi_h,u) \, {\mathrm{d}} u. \end{align}

Equation (3.4) is the standard formula for the canonical action coordinate, usually indicated with the symbol $I(E)$ for a given energy value $E$, and $q$ the canonical position (Arnol'd Reference Arnol'd2013). In (3.5), the identifications with canonical coordinates have been used, the constant energy value $E$ corresponds to the constant value of $\psi _h$ on $\varSigma _H$ and the expression (2.6) for the helical flux has been inverted to obtain $\psi _t(\psi _h,u)$. It is worth noting that on the right-hand side of this equation, the perturbed quantities appear, as $\psi _t$ and $\psi _h$, while on the left-hand side, we have the quantities through the island flux surfaces, identified by $\cdot _H$. As a side remark, the action coordinate in (3.4) is not related to the magnetic action in (2.1). The angle coordinate, defined on the helical axis, is defined by (Arnol'd Reference Arnol'd2013)

(3.6)\begin{align} {u_{H}} & = \int_{q_0}^q \frac{\partial p(I,q')}{\partial I} \, {\mathrm{d}} q' \end{align}

(3.7)\begin{align} & = \int_0^u \frac{\partial \psi_t({\psi_{H}},u')}{\partial {\psi_{{H} t}}} \, {\mathrm{d}} u' , \end{align}

which turns out to be the straight-helical-like angle defined on the helical axis (the $O$-point of the island) which increases by $2{\rm \pi}$ one turn around every helical flux surface.

Equation (3.5) implies that both magnetic fluxes (the action ${\psi _{{H} t}}$ and the Hamiltonian ${\psi _{H}}$ of the system) are constant of the motion, i.e. are constant on magnetic flux surfaces. Moreover, magnetic field lines written in the action-angle coordinates are straight lines in the (${u_{H}}, \varphi$) plane. The definitions of a helical angle and of a helical flux,

(3.8)\begin{gather} {u_{H}} = m {\vartheta_{H}} - n \varphi, \end{gather}

(3.9)\begin{gather}{\psi_{H}} = m {\psi_{{H} p}}-n {\psi_{{H} t}} , \end{gather}

bring to the definition of the new poloidal-like angle ${\vartheta _{H}}$ defined on the island $O$ point and, implicitly, of the poloidal flux through $\varSigma _H, {\psi _{{H} p}}=({\psi _{H}}+n {\psi _{{H} t}} )/m$. The rotational transform related to straight-field-lines in the plane (${\vartheta _{H}}, \varphi$) is defined by

(3.10)\begin{equation} {\iota_{H}}=\frac{{\mathrm{d}} {\vartheta_{H}}}{{\mathrm{d}} \varphi} = \frac{{\mathrm{d}} {\psi_{{H} p}}}{{\mathrm{d}} {\psi_{{H} t}}}, \end{equation}

which counts the poloidal and toroidal turns of an island magnetic field line seen by an external observer. Moreover, the Jacobian and the metric elements of the $({\psi _{H}},{\vartheta _{H}},\varphi )$ coordinate system allow us to calculate any geometric quantity related to the island domain, as the island volume, using the procedure that will be described in the next section for the stand-alone coordinate system.

We remark that the helical fluxes, $\psi _h(\psi _t^0,u)$ and ${\psi _{H}}({\psi _{{H} t}})$ in (2.6) and (3.9), respectively, are the same flux, due to the fact that the coordinate transformation $(\psi _t^0,u,\varphi )\mapsto ({\psi _{H}},{\vartheta _{H}},\varphi )$ is equivalent to a time-independent canonical transformation that does not change the Hamiltonian of the system. They differ just by the value of $\psi _h(\psi _t^0,u)$ on the helical axis, which ensures that ${\psi _{H}}, {\psi _{{H} t}}$ and ${\psi _{{H} p}}$ vanish there:

(3.11)\begin{equation} {\psi_{H}}=\psi_h(\psi_t^0,u)-\psi_h|_O, \end{equation}

where $\psi _h|_O=\psi _h(\psi _t^0,u)|_O$, whereas ${\psi _{H}}|_O=0$. Equation (3.11), together with (3.1) and (3.2) for $S_{O}$ and $S_{X}$, permits the interpretation of $\Phi _{{O}{X}}$ (which is the magnetic flux through the ribbon given by the $O$- and $X$-point orbits, as depicted in figure 1) as the helical flux through the separatrix surface. In fact,

(3.12)\begin{gather} - S_{O} = \psi_h|_O , \end{gather}

(3.13)\begin{gather}- S_{X} = \psi_h|_X = \psi_h|_O + {\psi_{H}}|_X, \end{gather}

simply yield

(3.14)\begin{equation} \Phi_{{O}{X}}= S_{O}-S_{X} ={-}{\psi_{H}}|_X.\end{equation}

Note that these considerations also apply to the next section.

4. Magnetic islands as stand-alone domains

Magnetic islands, even if embedded in a global toroidal magnetic field, can be regarded as separated plasma domains. In a previous work (Predebon et al. Reference Predebon, Momo, Suzuki and Auriemma2018), we described a method to characterize geometrically every isolated domain from discretized field maps. These maps are usually the outcome of a field-line tracing code or an MHD code: starting from a set of points in the usual cylindrical coordinates $(R,Z,\varphi )$ – with $R$ the distance from the axis of the torus, $Z$ the distance from the equatorial plane and $\varphi$ the geometrical toroidal angle – the method allows to obtain a magnetic coordinate system $({\psi _{{I} t}},{\vartheta _{I}},\varphi )$ in the island, with the following expression for the magnetic field $\boldsymbol {B}$:

(4.1)\begin{equation} \boldsymbol{B}=\frac{1}{2{\rm \pi}}(\boldsymbol{\nabla}{\psi_{{I} t}}\times\boldsymbol{\nabla}{\vartheta_{I}} +{\iota_{I}}({\psi_{{I} t}})\boldsymbol{\nabla}\varphi\times\boldsymbol{\nabla}{\psi_{{I} t}}), \end{equation}

where ${\iota _{I}}({\psi _{{I} t}})={\mathrm {d}}{\psi _{{I} p}}/{\mathrm {d}}{\psi _{{I} t}}$ is the rotational transform inside the island, $\varphi$ the toroidal angle and ${\vartheta _{I}}$ the poloidal angle such that the field lines are straight on the $({\vartheta _{I}},\varphi )$ plane, with $d{\vartheta _{I}}/d\varphi ={\iota _{I}}({\psi _{{I} t}})$. As $\varphi$ is the geometric toroidal angle, these coordinates are called symmetry flux coordinates (D'haeseleer et al. Reference D'haeseleer, Hitchon, Callen and Shohet1991). We remark that the $({\psi _{{I} t}},{\vartheta _{I}},\varphi )$ and $({\psi _{{H} t}},{\vartheta _{H}},\varphi )$ systems, being flux coordinates sharing the same toroidal angle, are mathematically equivalent. The indexes $\cdot _I$ and $\cdot _H$ denote the different procedure used to generate the respective flux coordinate systems.

Due to the high curvature of the surfaces in the proximity of the $X$-point, the method developed by Predebon et al. (Reference Predebon, Momo, Suzuki and Auriemma2018) does not allow to get an accurate description of the geometry in that region, thus we limit our reconstruction to the surface immediately preceding the separatrix. This is assumed to be the boundary of our domain.

There is freedom in the choice of the radial coordinate. Once a normalized radial coordinate ${\rho _{I}}$ is defined such that ${\rho _{I}}=0$ on the magnetic axis and ${\rho _{I}}=1$ on the last closed magnetic surface of the domain, and the Jacobian matrix ${\rm d}(R,Z,\varphi )/{\rm d}({\rho _{I}},{\vartheta _{I}},\varphi )$, or equivalently ${\rm d}(x,y,z)/{\rm d}({\rho _{I}},{\vartheta _{I}},\varphi )$, is known, we can calculate the (inverse) metric tensor $g^{ij}=\boldsymbol {\nabla } x^i\boldsymbol {\cdot }\boldsymbol {\nabla } x^j$ and the Jacobian $J=\sqrt {g}$, which for the coordinates $({\rho _{I}},{\vartheta _{I}},\varphi )$ will be explicitly written as $J_{I}$.

As the metric tensor is well defined in the whole island domain, we introduce here the observables that we intend to compare with the other approaches. Let us consider the width of the island itself. This can be measured with the ruler or can be calculated using the metric tensor. At a given toroidal angle $\varphi =\bar \varphi$, for a fixed poloidal angle ${\vartheta _{I}}=\bar {\vartheta _{I}}$, the (curvilinear) distance covered along the ${\rho _{I}}$ direction from the magnetic axis to a generic surface with ${\rho _{I}}\leq 1$ is

(4.2)\begin{equation} L|_{(\bar{\vartheta_{I}},\bar\varphi)}({\rho_{I}})=\int_0^{\rho_{I}} g_{{\rho_{I}}{\rho_{I}}}^{1/2}({\rho_{I}}',\bar{\vartheta_{I}},\bar\varphi)\,{\mathrm{d}}{\rho_{I}}', \end{equation}

where we have used the infinitesimal line element expression ${\mathrm {d}} l^2=g_{ij}\,{\mathrm {d}} x^i\,{\mathrm {d}} x^j$ restricted to the radial direction. In figure 3, we show the $\varphi =0$ section of a $(1,1)$ island in a circular tokamak based on the RFX-mod geometry, obtained again with the field line tracing code Flit. At this section, the ${\vartheta _{I}}=0$ and ${\vartheta _{I}}={\rm \pi}$ coordinate lines correspond to the horizontal cut of the island, so that the island width is simply given by

(4.3)\begin{equation} {W_{I}}|_{\bar\varphi=0} = L|_{(0,0)}(1) + L|_{({\rm \pi},0)}(1). \end{equation}

Other useful quantities for a comparison with the other approaches include the volume enclosed by the surface with radius ${\rho _{I}}$,

(4.4)\begin{equation} {V_{I}}({\rho_{I}}) = \int_{[0,{\rho_{I}}]\times[0,2{\rm \pi}]\times[0,2{\rm \pi}]} J_{I}\,{\mathrm{d}}{\rho_{I}}'\,{\mathrm{d}}{\vartheta_{I}}\,{\mathrm{d}}\varphi, \end{equation}

as well as the poloidal (the ribbon poloidal flux, as it is called by D'haeseleer et al. (Reference D'haeseleer, Hitchon, Callen and Shohet1991)) flux,

(4.5)\begin{equation} {\psi_{{I} p}}({\rho_{I}}) = \frac{1}{2{\rm \pi}}\int_{[0,{\rho_{I}}]\times[0,2{\rm \pi}]\times[0,2{\rm \pi}]} B^{{\vartheta_{I}}}\,J_{I} \,{\mathrm{d}}{\rho_{I}}'\,{\mathrm{d}}{\vartheta_{I}}\,{\mathrm{d}}\varphi, \end{equation}

and the toroidal flux,

(4.6)\begin{equation} {\psi_{{I} t}}({\rho_{I}}) = \frac{1}{2{\rm \pi}}\int_{[0,{\rho_{I}}]\times[0,2{\rm \pi}]\times[0,2{\rm \pi}]} B^{\varphi}\,J_{I} \,{\mathrm{d}} {\rho_{I}}'\,{\mathrm{d}}{\vartheta_{I}}\,{\mathrm{d}}\varphi, \end{equation}

where the poloidal and toroidal contravariant components of the field are given by $B^{\vartheta _{I}}=\boldsymbol {B}\boldsymbol {\cdot }\boldsymbol {\nabla }{\vartheta _{I}}$ and $B^\varphi =\boldsymbol {B}\boldsymbol {\cdot }\boldsymbol {\nabla }\varphi$, respectively.

Figure 3. $\varphi =0$ discretized field map of a (1,1) island in a circular tokamak (top half of the section) and corresponding flux-coordinate grid (bottom half) with the ${\rho _{I}}=$ const. lines (in blue, thick for ${\rho _{I}}=1$) and the ${\vartheta _{I}}=k{\rm \pi} /8, k$ integer lines (in purple, thick for ${\vartheta _{I}}=0,{\rm \pi}$).

Combining (4.5) and (4.6) to define the helical flux ${\psi _{{I} {H}}} = m {\psi _{{I} p}} - n {\psi _{{I} t}}$, the island width can be again inferred from (2.11) with $\Phi _{{O}{X}}=-{\psi _{{I} {H}}}|_X$, as in (3.14), since the helical flux on the island axis vanishes identically. In this case, the $\Phi _{{O}{X}}$ flux is calculated at ${\rho _{I}}=1$, not at the separatrix, as a line integral following one of the tips of the island, which is the curve that best approximates the separatrix, corresponding to the angle $u_I=m{\vartheta _{I}}-n\varphi =\pm {\rm \pi}/2$.

5. A comparison of the different approaches

In the following, we provide a comparison of some observables using the different approaches described above. For this comparison, we consider the islands already introduced in the previous sections, namely a (1,1) island in a circular tokamak and a (1,7) island in the RFP, both based on RFX-mod, a circular toroidal device with major radius $R_0=2$ m and minor radius $a=0.46$ m which can perform operation in either configuration (Sonato et al. Reference Sonato, Chitarin, Zaccaria, Gnesotto, Ortolani, Buffa, Bagatin, Baker, Dal Bello, Fiorentin, Grando, Marchiori, Marcuzzi, Masiello, Peruzzo, Pomaro and Serianni2003).

The reconstruction of the magnetic island topology is based on the calculation of the helical perturbations to the zeroth-order equilibrium. According to the procedure developed by Zanca & Terranova (Reference Zanca and Terranova2004), this is done solving a Newcomb-like equation, i.e. the linearized force-balance equation with the linearized Ampere's law ensuring a divergence-free magnetic field, using the external magnetic measurements as boundary conditions. Due to the set of measurements available in RFX-mod (48 toroidal arrays of four poloidally equispaced probes for both the radial and toroidal field components), a rich spectrum of Fourier components can be reconstructed for the magnetic perturbations. In this work, we focus on the dominant resonant Fourier component generating the $(m,n)$ island, as specified in (2.3) and (2.4). The mode is $(m,n)=(1,1)$ for the tokamak case, pulse 38 818 at $t=362$ ms and $(m,n)=(1,7)$ for the RFP case, which is the averaged discharge described by Momo et al. (Reference Momo, Isliker, Cavazzana, Zuin, Cordaro, Bruna, Martines, Predebon, Rea, Spolaore, Vlahos and Zanca2020). The resulting islands have been characterized geometrically by Predebon et al. (Reference Predebon, Momo, Suzuki and Auriemma2018), where the discretized field maps have been produced by the field line tracing code Flit (Innocente et al. Reference Innocente, Lorenzini, Terranova and Zanca2017) which is indeed based on Newcomb's perturbed fluxes as input.

As already mentioned, the comparison with the stand-alone approach of § 4 is possible only within the surface ${\rho _{I}}=1$, which for both the RFP and tokamak islands is the last surface of the Poincaré section before the separatrix. For the other approaches, the radial domain extends from the $O$-point of the island to the separatrix.

In figure 4 and table 1, we summarize the results of the comparison. In the figure, the toroidal flux, the rotational transform and the volume profiles are plotted as a function of the poloidal flux for the approaches of § 3 (red lines) and § 4 (black). In particular, the poloidal and toroidal fluxes and the rotational transform defining the island flux surfaces are computed from (3.5) and (3.10) in the approach of § 3, and from (4.5) and (4.6) in the approach of § 4. The volume is computed in both cases from (4.4), using the related coordinate system and Jacobian. The comparison can be considered satisfactory. The small difference in the reconstruction of the rotational transform profile (approximately 0.4 % at the surface ${\rho _{I}}=1$ for the tokamak, 0.5 % for the RFP) explains the pointwise differences in the fluxes that appear in the following table.

Figure 4. (a,b) Toroidal flux, (c,d) $\iota$ and (e,f) volume as a function of the poloidal flux for (a,c,e) a (1,1) tokamak island and (b,d,f) a (1,7) RFP island, for the approaches of § 3 (red lines) and § 4 (black).

Table 1. For a (1,1) tokamak island and a (1,7) RFP island, $\cdot ^{\dagger}$ quantities refer to the perturbative method of § 3, $\cdot ^\ddagger$ quantities to the non-perturbative method of § 4, $\cdot ^\circ$ quantities to the perturbative method of § 2; in the second row, $W_{\Phi _{{O}{X}}}$ is calculated from $\Phi _{{O}{X}}$ by means of (2.11) for the three different methods: $\Phi _{{O}{X}}\,^{\dagger} =m{\psi _{{H} p}}-n{\psi _{{H} t}}, \Phi _{{O}{X}}\,^\ddagger =m{\psi _{{I} p}}-n{\psi _{{I} t}}, \Phi _{{O}{X}}\,^\circ$ as defined in (2.10). $W_I|_{\bar \varphi =0}$ is the width as defined in (4.3). $W_{r}|_{\bar \varphi =0}$ is the width measured with the ruler at $\varphi =0$.

For the two islands, in the table, we review the most relevant quantities calculated at the $X$-point and at the last surface at the Poincaré map (${\rho _{I}}=1$). For the perturbative method of § 3 applied to the ${\rho _{I}}=1$ surface, we assume ${\psi _{{H} p}}|_{{\rho _{I}}=1}={\psi _{{I} p}}|_{{\rho _{I}}=1}$ and derive the other quantities based on this reference value. The island volume, and the toroidal and poloidal fluxes through the island are reported in the first block of the table. In the second block, the $\Phi _{{O}{X}}$ flux is computed from (2.10) for the approach of § 2, as the helical flux through the separatrix using (3.14) for the approach of § 3, and as the path integral along the (best approximation of the) $X$-point identified by the angle $u_I=m{\vartheta _{I}}-n\varphi =\pm {\rm \pi}/2$ in the geometric approach of § 4 (proving also that $\Phi _{{O}{X}}=-S_{X}$ when the fluxes vanish on the $O$-point). Then, for the three methods, the related island width is calculated with (2.11). To complete the table, we also report the island width as resulting from (4.3) and as measured with a ruler from the Poincaré maps at $\varphi =0, W_{r}|_{\bar \varphi =0}$.

The method which best estimates $W_{r}|_{\bar \varphi =0}$ is that of § 4, which is not surprising as the island geometry is directly derived from the Poincaré map itself: the ${W_{I}}|_{\bar \varphi =0}$ width from (4.3) perfectly matches $W_{r}|_{\bar \varphi =0}$. From the $\Phi _{{O}{X}}$ flux calculated as path integral along one of the tips of the island, we provide the width $W_{\Phi _{{O}{X}}}$ from (2.11), which yields a value within $1\,\%$ and $5\,\%$ error with respect to $W_{r}|_{\bar \varphi =0}$, for tokamak and RFP, respectively.

However, the perturbative methods of §§ 2 and 3 overestimate the total island width, within $6\,\%$ and $10\,\%$ error with respect to $W_{r}|_{\bar \varphi =0}$, for tokamak and RFP case, respectively. We recall that the formula of (2.11) for $W_{\Phi _{{O}{X}}}$ comes from the formal analogy between the typical textbook deformation of the flux surfaces around the resonance due to the opening of a magnetic island and the phase diagram of a pendulum, valid for small perturbations. This explains the error introduced by the application of this formula to the specific cases shown here, where the perturbation cannot be considered small. Large perturbations cause a displacement of the $O$-point and $X$-point from the rational surface where the radial derivatives are calculated. However, the correction to $\Phi _{{O}{X}}$ when calculated as the helical flux through the island separatrix (method of § 3), with respect to its estimate based on the axisymmetric equilibrium (method of § 2), slightly improves the evaluation of the width.

6. Summary

We have considered three different approaches for a geometric characterization of magnetic islands: (i) perturbative; (ii) perturbative but leading to the definition of a new single-helicity equilibrium; and (iii) non-perturbative based on the availability of a discretized field map.

In the first perturbative method, the island is described in a Hamiltonian context using the definition of action $S_\gamma$ of a magnetic system. The novelty of this approach, first introduced by Escande & Momo (Reference Escande and Momo2024), is the definition of a coordinate-independent flux with a clear geometrical meaning – here named $\Phi _{{O}{X}}$ – related to the island width through (2.11). This formula, coming from the analogy between a textbook magnetic island and the cat's eye shape of the phase diagram of a pendulum, is very similar to the classical formulae for the island width (Hazeltine & Meiss Reference Hazeltine and Meiss1992; Park et al. Reference Park, Boozer and Menard2008), but with a new meaning of the flux that appears in all other formulations. The second method, developed by Martines et al. (Reference Martines, Lorenzini, Momo, Terranova, Zanca, Alfier, Bonomo, Canton, Fassina, Franz and Innocente2011) and Momo et al. (Reference Momo, Martines, Escande and Gobbin2011) for the RFP equilibrium and here extended to the tokamak configuration, starts from a perturbative approach that sums the zeroth-order equilibrium and the single resonant perturbation generating the island, then leads to the definition of a new single-helicity equilibrium using definitions specific to Hamiltonian mechanics. The third, non-perturbative method, is based on the availability of a discretized field map and defines the island domain through its geometrical definitions, as first developed by Predebon et al. (Reference Predebon, Momo, Suzuki and Auriemma2018).

A detailed comparison has been carried out among the three methods applying them to two experimental islands of RFX-mod, both tokamak and RFP. We have provided an estimate of some observables showing that they are, in general, in good agreement with each other. As a novelty, moreover, we have extended the use of the expression (2.11) for the island width in a broader context than the perturbative approach in which it was first developed, thanks to a more comprehensive geometric interpretation of the $\Phi _{{O}{X}}$ flux. In particular, the $\Phi _{{O}{X}}$ flux, originally defined as the flux through the ribbon defined by the $O$- and $X$-point orbits, is identified as the helical flux through the island separatrix.

As a final remark, the (third) non-perturbative method, strictly based on the geometry of the flux surfaces, is the one which provides the most accurate description of the island observables for a large part of the island domain, failing, however, to describe the separatrix due to the high curvature of the flux surfaces in the neighbourhood of the $X$-point. This approach applies to any isolated region of the plasma, without any assumptions on the symmetry of the system and the possible interactions with other perturbations with different helicities. However, the two perturbative methods provide a geometric characterization which is in reasonable agreement with the non-perturbative method, at least in the cases with a strong helical symmetry. Easier to apply, based as they are on a single-mode perturbation from a linear Newcomb-like analysis, these methods can satisfactorily highlight the most relevant features of an island without the need to build discretized field maps and/or use MHD codes, providing a viable method for a fast description of a magnetic island domain.